Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,13)(14,17,18,16)(19,20,23,24)(21,25,27,31)(22,26,30,32)(28,33,29,34);(4,7)(8,9)(11,13,15,12)(14,17,18,16)(20,24)(21,27)(26,32)(28,29);(11,14,15,18)(12,16,13,17)(19,21)(20,25)(22,29)(23,27)(24,31)(26,34)(28,30)(32,33);(3,4)(5,8)(6,7)(9,10)(19,22,23,30)(20,26,24,32)(21,28,27,29)(25,33,31,34);(3,5)(4,8)(6,10)(7,9)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,34)(30,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613726);